Problem: Rewrite as a simplified fraction. ${1.32\overline{1} = {?}}$
Answer: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}1000x &= 1321.1111...\\ 100x &= 132.1111...\end{align*}}$ Subtracting the two equations, we have: ${900x = 1189}$ ${x = \dfrac{1189}{900}} \text{or} {1 \dfrac{289}{900}}$